Consistent Truncation Research Articles

Charting the conformal manifold of holographic CFT$_2$'s

We construct new continuous families of AdS_3× S^3× T^4AdS3×S3×T4 and AdS_3× S^3× S^3× S^1AdS3×S3×S3×S1 solutions in heterotic and type II supergravities. These families are found in three-dimensional consistent truncations and controlled by 17 parameters, which include TsT \betaβ deformations and encompass several supersymmetric sub-families. The different uplifts are constructed in a unified fashion by means of Exceptional Field Theory (ExFT). This allows the computation of the Kaluza-Klein spectra around the deformations, to test the stability of the solutions, and to interpret them holographically and as worldsheet models. To achieve this, we describe how the half-maximal S(8,8)S(8,8) ExFT can be embedded into E_{8(8)}E8(8) ExFT.

The geometry of large charge multi-traces in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM

We construct a one-parameter family of half-BPS solutions of type IIB supergravity using a consistent truncation to gauged five-dimensional supergravity. For small values of the parameter, the solution reduces to the linear perturbation of AdS5 × S5 dual to the chiral primary operator in the stress-tensor multiplet, and we give evidence that the geometry is regular and asymptotes AdS in a normalisable way for arbitrarily large values of the parameter. We conjecture that the solution is the gravity dual of a “heavy” multi-trace operator in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SU(N) Super Yang-Mills made by p copies of the stress-tensor chiral primary operator, with p of order N2 in the large N limit. We perform some holographic checks supporting this duality map.

Infinite and finite consistent truncations on deformed generalised parallelisations

Given a manifold M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathbbm{M} $$\\end{document} admitting a maximally supersymmetric consistent truncation, we show how to formulate new consistent truncations by restricting to a set of Kaluza-Klein modes on M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathbbm{M} $$\\end{document} invariant under some subgroup of the group of isometries of M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathbbm{M} $$\\end{document}. These truncations may involve either finite or infinite sets of modes. We provide their global description using exceptional generalised geometry to construct a ‘deformed’ generalised parallelisation starting with that on M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathbbm{M} $$\\end{document}. This allows us to explicitly embed known consistent truncations directly into exceptional generalised geometry/exceptional field theory, and to obtain the equations governing situations where the consistent truncation retains an infinite tower of modes.

Double copy of 3D Chern-Simons theory and 6D Kodaira-Spencer gravity

We apply an algebraic double copy construction of gravity from gauge theory to three-dimensional (3D) Chern-Simons theory. The kinematic algebra K is the 3D de Rham complex of forms equipped, for a choice of metric, with a graded Lie algebra that is equivalent to the Schouten-Nijenhuis bracket on polyvector fields. The double copied gravity is defined on a subspace of K⊗K¯ and yields a topological double field theory for a generalized metric perturbation and two 2-forms. This local and gauge invariant theory is non-Lagrangian but can be rendered Lagrangian by abandoning locality. Upon fixing a gauge this reduces to the double copy of Chern-Simons theory previously proposed by Ben-Shahar and Johansson. Furthermore, using complex coordinates in C3 this theory is related to six-dimensional (6D) Kodaira-Spencer gravity in that truncating the two 2-forms and one equation yields the Kodaira-Spencer equations on a 3D real slice of C3. The full 6D Kodaira-Spencer theory can instead be obtained as a consistent truncation of a chiral double copy. Published by the American Physical Society 2024

SCFT deformations via uplifted solitons

A holographic method for implementing a particular supersymmetry-preserving deformation to 4d SCFTs is presented. At the heart of the procedure is a soliton solution of minimal d=5 gauged supergravity. Embedding this solution into ten- and eleven-dimensional string theory backgrounds of the form AdS×5M, we systematically construct a range of new solutions. Each holographically realizes a twisted compactification of the SCFT4 dual to the original background. In the IR, the resulting SQFTs flow to gapped three-dimensional systems. Using a variety of holographic observables, we give evidence for this interpretation and for confinement in the deformed SQFTs. Our method applies to any holographic solutions admitting a consistent truncation to minimal d=5 gauged supergravity, and can likely be generalized to solutions with other AdSd factors.

Generalized dualities for heterotic and type I strings

We define generalized dualities for heterotic and type I strings based on consistent truncations to half-maximal gauged supergravities in more than three dimensions. The latter are constructed from a generalized Scherk-Schwarz ansatz in heterotic double field theory that satisfies the strong constraint. Necessary and sufficient conditions on the resulting embedding tensor are discussed, showing that only certain gaugings, called geometric, can arise from this procedure. For all of them, we explicitly construct the internal geometry and gauge potentials. In general, this construction is not unique and permits different uplifts which are used to define generalized T-duality. Two examples are worked out underlying the utility of our approach to explore new dualities and uplifts of half-maximal gauged supergravities.

A story of non-conformal branes: spindles, disks, circles and black holes

We consider the (p + 2)-dimensional gauged supergravities arising as a consistent truncation of type II on S8−p, which are associated with the near-horizon limit of Dp-branes, for p = 2, 4, 5, 6 (and NS5-branes for p = 5). In a truncation of these theories with only abelian gauge fields and scalars, we find several classes of new solutions, with and without supersymmetry. Our ansatz for such backgrounds is inspired by the recent progress in the study of branes wrapped on orbifolds, but unlike those examples we consider “non-conformal branes”, i.e. no Anti de Sitter factors in the metric. Focusing on cases with non-trivial gauge fields, we can divide the solutions that we present in three categories: 1) branes wrapping Riemann surfaces, spindles and disks, 2) branes wrapped on a circle with a holonomy for the gauge field along the circle and 3) electrically charged black holes in gauged supergravity, which uplift to rotating branes in ten dimensions. We carefully analyze the conditions for supersymmetry in all these cases.

Bound states from the spectral Bethe-Salpeter equation

We compute the bound state properties of three-dimensional scalar ϕ4 theory in the broken phase. To this end, we extend the recently developed technique of spectral Dyson-Schwinger equations to solve the Bethe-Salpeter equation and determine the bound state spectrum. We employ consistent truncations for the two-, three- and four-point functions of the theory that recover the scaling properties in the infinite coupling limit. Our result for the mass of the lowest-lying bound state in this limit agrees very well with lattice determinations. Published by the American Physical Society 2024

Bulk-boundary thermodynamics of charged black holes in higher-derivative theory

The connection between bulk and boundary thermodynamics in Einstein–Maxwell theory is well established using AdS/CFT correspondence. In the context of general higher-derivative gravity coupled to a U(1) gauge field, we examine the resemblance of the first law of thermodynamics between bulk and boundary, followed by an extended phase space description on both sides. Higher-derivative terms related to different powers of the string theory parameter α′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha '$$\\end{document} emerged from a consistent truncation in the bulk supergravity action. We demonstrate that one must include the fluctuation of α′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha '$$\\end{document} in the bulk thermodynamics as a bookkeeping tool to match the bulk first law and Smarr relation with the boundary side. Consequently, the Euler relation and the boundary first law are altered by adding two central charges (a,c).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\ exttt{a}}, {\ exttt{c}}).$$\\end{document} To support our general conclusion, we consider the black hole in Gauss–Bonnet gravity and the general four-derivative theory. Finally, we examine the bulk and boundary aspects of the extended phase space description for higher-derivative-corrected black holes.

Hierarchy of curvatures in exceptional geometry

Despite remarkable success in describing supergravity reductions and backgrounds, generalized geometry and exceptional field theory are still lacking a fundamental object of differential geometry, the Riemann tensor. We show that to construct it, a hierarchy of connections is required. They complement the spin connection with higher representations known from the tensor hierarchy. This approach allows to define generalized homogeneous spaces which underlie generalized U-duality, admit consistent truncations and provide a huge class of new flux backgrounds with nontrivial structure groups. Published by the American Physical Society 2024

Vacua scan of 5d, [formula omitted] consistent truncations

In this letter we present a scan for new vacua within consistent truncations of eleven/ten-dimensional supergravity down to five dimensions that preserve N=2 supersymmetry, after their complete classification in [1]. We first make explicit the link between the equations of exceptional Sasaki-Einstein backgrounds in [2] and the standard BPS equations for 5d N=2 of [3]. This derivation allows us to expedite a scan for vacua preserving N=2 supersymmetry within the framework used for the classification presented in [1].

Supersymmetric Wormholes in String Theory.

We construct a large family of Euclidean supersymmetric wormhole solutions of type IIB supergravity which are asymptotically AdS_{5}×S^{5}. The solutions are constructed using consistent truncation to maximally gauged supergravity in five dimensions which is further truncated to a four scalar model. Within this model we perform a full analytic classification of supersymmetric domain wall solutions with flat Euclidean domain wall slices. On each side of the wormhole, the solution asymptotes to AdS_{5} dual to N=4 supersymmetric Yang-Mills deformed by a supersymmetric mass term.

Y-algebroids and E7(7)× ℝ+-generalised geometry

We define the notion of Y-algebroids, generalising the Lie, Courant, and exceptional algebroids that have been used to capture the local symmetry structure of type II string theory and M-theory compactifications to D ≥ 5 dimensions. Instead of an invariant inner product, or its generalisation arising in exceptional algebroids, Y-algebroids are built around a specific type of tensor, denoted Y , that provides exactly the necessary properties to also describe compactifications to D = 4 dimensions. We classify “M-exact” E7-algebroids and show that this precisely matches the form of the generalised tangent space of E7(7) × ℝ+-generalised geometry, with possible twists due to 1-, 4- and 7-form fluxes, corresponding physically to the derivative of the warp factor and the M-theory fluxes. We translate the notion of generalised Leibniz parallelisable spaces, relevant to consistent truncations, into this language, where they are mapped to so-called exceptional Manin pairs. We also show how to understand Poisson-Lie U-duality and exceptional complex structures using Y-algebroids.

Rounding out the story of higher derivative consistent truncations

At the two-derivative order, the group manifold reduction of heterotic supergravity on S3 results in a half-maximal 7D gauged supergravity coupled to three vector multiplets, and a further truncation can be taken to remove the vector multiplets. We demonstrate that this truncation remains consistent at the four-derivative level; we do so both by analysis of the equations of motion and the supersymmetry variations.

Higgs-like (pseudo)scalars in AdS4, marginal and irrelevant deformations in CFT3, and instantons on S 3

Employing a 4-form ansatz of 11-dimensional supergravity over a non-dynamical background and setting the internal space as an Hopf fibration on , we obtain a consistent truncation. The (pseudo)scalars, in the resulting scalar equations in Euclidean AdS space, may be considered to arise from (anti)M-branes wrapping around the internal directions in the (Wick-rotated) skew-whiffed M2-brane background (as the resulting theory is for anti-M2-branes), thus realizing the modes after swapping the three fundamental representations , , and of . Taking the backreaction on the external and internal spaces, we obtain the massless and massive modes, corresponding to exactly marginal and marginally irrelevant deformations on the boundary CFT , respectively. Subsequently, we obtain a closed solution for the bulk equation and compute its correction with respect to the background action. Next, considering the Higgs-like (breathing) mode , having all supersymmetries as well as parity and scale-invariance broken, solving the associated bulk equation with mathematical methods, specifically the Adomian decomposition method, and analyzing the behavior near the boundary of the solutions, we realize the boundary duals in the -singlet sectors of the ABJM model. Then, introducing the new dual deformation operators made of bi-fundamental scalars, fermions, and gauge fields, we obtain the -invariant solutions as small instantons on a three-sphere with the radius at infinity, which correspond to collapsing bulk bubbles leading to big-crunch singularities.

Symmetry breaking and consistent truncations from M5-branes wrapping a disc

We construct new supersymmetric solutions corresponding to M5-branes wrapped on a topological disc by turning on additional scalars in the background. The presence of such scalar fields breaks one of the U(1) isometries of the internal space, explicitly realising the breaking by the Stückelberg mechanism observed previously. In addition, we construct a consistent truncation of maximal seven-dimensional gauged supergravity on the disc to five-dimensional Romans’ SU(2) × U(1) gauged supergravity, allowing us to construct a plethora of new supergravity solutions corresponding to more general states in the dual SCFTs as well as solutions corresponding to M5-branes wrapping four-dimensional orbifolds.

Maximal D = 2 supergravities from higher dimensions

We develop in detail the general framework of consistent Kaluza-Klein truncations from D=11 and type II supergravities to gauged maximal supergravities in two dimensions. In particular, we unveil the complete bosonic dynamics of all gauged maximal supergravities that admit a geometric uplift. Our construction relies on generalised Scherk-Schwarz reductions of E9 exceptional field theory. The application to the reduction of D=11 supergravity on S8 × S1 to SO(9) gauged supergravity is presented in a companion paper.

Consistent truncation of eleven-dimensional supergravity on S8× S1

Eleven-dimensional supergravity on S8 × S1 is conjectured to be dual to the M-theory matrix model. We prove that the dynamics of a subset of fluctuations around this background is consistently described by D = 2 SO(9) gauged maximal supergravity. We provide the full non-linear uplift formulæ for all bosonic fields. We also present a further truncation to the SO(3)×SO(6) invariant sector and discuss its relation to the BMN matrix model at finite temperature. The construction relies on the framework of generalised Scherk-Schwarz reductions, established for E9 exceptional field theory in a companion paper. As a by-product, we severely constrain the most general gauge deformations in D = 2 admitting an uplift to higher dimensions.

Adding fluxes to consistent truncations: IIB supergravity on AdS3× S3× S3× S1

We use E8(8) Exceptional Field Theory to construct the consistent truncation of IIB supergravity on S3× S3× S1 to maximal 3-dimensional mathcal{N} = 16 gauged supergravity containing the mathcal{N} = (4, 4) AdS3 vacuum. We explain how to achieve this by adding a 7-form flux to the S1 reduction of the dyonic E7(7) truncation on S3× S3 previously constructed in the literature. Our truncation Ansatz includes, in addition to the mathcal{N} = (4, 4) vacuum, a host of moduli breaking some or all of the supersymmetries. We explicitly construct the uplift of a subset of these to construct new supersymmetric and non-supersymmetric AdS3 vacua of IIB string theory, which include a range of perturbatively stable non-supersymmetric 10-d vacua. Moreover, we show how the supersymmetric direction of the moduli space of AdS3 vacua of six-dimensional gauged supergravity studied in [1] is compactified upon lifting to 10 dimensions, and find evidence of T-duality playing a role in global aspects of the moduli space. Along the way, we also derive the form of 3-dimensional mathcal{N} = 16 gauged supergravity in terms of the embedding tensor and rule out a 10-/11-dimensional origin of some 3-dimensional gauged supergravities.

Microstrata

Microstrata are the non-extremal analogues of superstrata: they are smooth, non-extremal (non-BPS) solitonic solutions to IIB supergravity whose deep-throat limits approximate black holes. Using perturbation theory and numerical methods, we construct families of solutions using a consistent truncation to three-dimensional supergravity. The most general families presented here involve two continuous parameters, or amplitudes, and four quantized parameters that set the angular momenta and energy levels. Our solutions are asymptotic to the vacuum of the D1-D5 system: AdS3 × S3 × \U0001d54b4. Using holography, we show that the they are dual to multi-particle states in the D1-D5 CFT involving a large number of mutually non-BPS supergravitons and we determine the anomalous dimensions of these states from the binding energies in supergravity. These binding energies are uniformly negative and depend non-linearly on the amplitudes of the states. In one family of solutions, smoothness restricts some of the fields to lie on a special locus of the parameter space. Using precision holography we show that this special locus can be identified with the multi-particle states constructed via the standard OPE of the single-particle constituents. Our numerical analysis shows that microstrata are robust at large amplitudes and the solutions can be obtained to very high precision.